Problem: Determine how many solutions exist for the system of equations. ${4x+y = 8}$ ${y = -4x+8}$
Convert both equations to slope-intercept form: ${4x+y = 8}$ $4x{-4x} + y = 8{-4x}$ $y = 8-4x$ ${y = -4x+8}$ ${y = -4x+8}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -4x+8}$ ${y = -4x+8}$ Both equations have the same slope and the same y-intercept, which means the lines would completely overlap. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Since any solution of ${4x+y = 8}$ is also a solution of ${y = -4x+8}$, there are infinitely many solutions.